Pricing Ratchet EIA under Heston’s Stochastic Volatility with Deterministic Interest

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Pricing Ratchet EIA under Heston’s Stochastic Volatility with

Deterministic Interest

Dezhao Han

A Thesis

in

The Department

of

Mathematics and Statistics

Presented in Partial Fulfillment of the Requirements

for the Degree of Master of Science (Mathematics) at

Concordia University

Montreal, Quebec, Canada

August 2012

Dezhao Han, 2012

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CONCORDIA UNIVERSITY

School of Graduate Studies

This is to certify that the thesis prepared

By: Dezhao Han

Entitled: Pricing Ratchet EIA under Heston’s Stochastic Volatility with Deterministic Interest

and submitted in partial fulfillment of the requirements for the degree of

Master of Science (Mathematics)

complies with the regulations of the University and meets the accepted standards with respect to originality and quality.

Signed by the final Examining Committee:

Examiner Dr. J. Garrido Examiner Dr. C. Hyndman Thesis Supervisor Dr. P. Gaillardetz Approved by

Chair of Department or Graduate Program Director

2012

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ABSTRACT

Pricing Ratchet EIA under Heston’s Stochastic Volatility with Deterministic Interest

Dezhao Han

Since its introduction in 1995, equity-indexed annuities (EIAs) received increasing attention from investors. Most of the pricing and hedging for different types of EIAs have been obtained in the Black-Scholes (BS) framework. In this framework the un-derlying asset is assumed to follow a geometric Brownian motion. However, the BS model is plagued by its assumption of constant volatility, while stochastic volatility models have become increasingly popular. In this paper we assume that the asset price follows Heston’s stochastic volatility model with deterministic interest, and introduce two methods to price the ratchet EIA.

The first method is called the joint transition probability density function (JT-PDF) method. Given the JTPDF of the asset price and variance, pricing ratchet EIAs boils down to a question of solving multiple integrals. Here, the multiple integral is solved using Quasi Monte Carlo methods and the importance sampling technique. The other method used to evaluate EIAs prices is called the conditional expectation (CE) approach. Conditioning on the volatility path, we first price the rachet EIA analytically in a BS framework. Then the price in Heston framework can be obtained by simulating the volatility path. Greeks for the ratchet EIA can also be calculated by the JTPDF and CE methods. At the end, we carry out some sensitivity analyses for ratchet EIAs’ prices and Greeks.

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Acknowledgements

I would like to express my special thanks and gratitude to my supervisor Dr. Patrice Gaillardetz, who chose me such an interesting topic and provided me with constant encouragement and inspiration throughout my academic research. I would also like to thank my co-supervisor, Professor Jos´e Garrido, for all his help through-out my studies at Concordia. Many thanks to Dr. Xiaowen Zhou, who makes me feel at home.

Finally, I would like to thank my families, specially my parents, and also my friends. All of you make me brave for life.

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Contents

1 Economic Model 4

1.1 From Black-Scholes to Stochastic Volatility . . . 4

1.2 Heston’s Stochastic Volatility . . . 6

1.3 Deterministic Interest Rate. . . 7

1.4 Non-Arbitrage Assumption and Heston Framework . . . 8

1.5 European Call Price . . . 10

1.5.1 Characteristic Functions . . . 11

1.5.2 Heston PDE . . . 17

1.5.3 Solving the Characteristic Functions . . . 19

1.6 Calibrating the Heston Model . . . 24

2 Joint Transition PDF of Heston Model 29 2.1 The Joint Transition Probability Density Function . . . 30

2.2 The Quadrature of The TPDF Integral . . . 32

2.3 Pricing European Path-Dependent Derivatives using TPDF . . . 36

2.4 Importance Sampling . . . 37

2.4.1 Importance Sampling . . . 37

2.4.2 JTPDF Approximation . . . 38

2.5 Generate ˜pSamples using Quasi-Monte Carlo Method . . . 40

2.5.1 Quasi-Monte Carlo Method . . . 40

2.5.2 Generating Samples From ˜p . . . 41

3 The Conditional Probability Density Function of Asset Prices 43 3.1 Pricing Derivatives via Conditional Expectation . . . 43

3.2 Conditional Density of the Asset Price . . . 43

3.3 Simulate the Path for the Stochastic Variance and the Integrated Variance 45 3.4 Comparison of the JTPDF and CE Methods . . . 46

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4 Application to Ratchet EIA 48

4.1 Investment Guarantees . . . 48

4.2 Introduction to EIAs . . . 49

4.3 Pricing EIAs. . . 51

4.3.1 Pricing Ratchet EIAs via the TPDF Method. . . 51

4.3.2 Pricing Ratchet EIAs via the CE Method. . . 52

4.4 Greeks of the Ratchet EIA . . . 55

5 Numerical Results 58 5.1 European Call Option . . . 58

5.2 Ratchet EIAs . . . 59 5.2.1 Prices . . . 59 5.2.2 Fair Values . . . 60 5.2.3 Greeks . . . 62 6 Conclusions 66 A Appendix A 74 B Appendix B 79 C Appendix C 81

D Appendix D: Bilinear Interpolation 84

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List of Figures

1 Calibration result for NSS model. . . 26

2 Calibration result for Heston parameters. . . 28

3 Figure of the transition PDF . . . 31

4 f(x) =x2cos(60x), x[3,3] . . . . 33

5 Pseudo-random numbers vs. Sobol’s sequence, 5000-by-5000 . . . 41

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List of Tables

1 Stochastic volatility models . . . 5

2 Yield curve in percentage . . . 25

3 NSS model calibration . . . 25

4 Heston parameters under risk neutral measure . . . 27

5 Greeks . . . 55

6 European call prices . . . 58

7 Applying the CE approach to European call options . . . 59

8 Price of the 7-year ratchet EIA . . . 60

9 Fair values (in percentage) given by the QE method . . . 61

10 Errors given by the JTPDF approach . . . 61

11 Errors given by the CE approach . . . 62

12 ΛRat0 of ratchet EIAs . . . 63

13 ΛRat0 of ratchet EIA (II) . . . 64

14 ΔRat 1.5 of ratchet EIA . . . 65

15 ΓRat 1.5 of ratchet EIA . . . 65

16 ΔRat 1.5 of ratchet EIA (II) . . . 65

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Introduction

An equity-indexed annuity (EIA) is an innovative hybrid insurance product which pro-vides participation in the financial market. The EIA’s return is linked to the performance of an equity index, typically the S&P 500 index, while a minimal return is guaranteed on its initial investment. Investors sacrifice some potential upside return for the downside protection, which means that the policy earns a non-negative return even when the mar-ket performs poorly. EIAs are increasingly popular since their introduction in 1995 by Keyport Life Insurance, and according to Marrion et al. (2010), EIAs’ sales have grown dramatically from $3.00 billion in 1997 to $30.2 billion in 2009. In 2008 the sales of fixed-indexed annuities, the generalization of EIAs, represented 42% of the annuities sold by agents, but in 2010 their market share shrank to 25% (see soa.org). This could be explained by the fact that the companies failed to hedge EIAs during the financial crisis and they became reluctant to issue such contracts. Thus pricing and hedging EIAs are interesting topics.

Designs of EIAs vary according to the companies that sell them. The simplest EIA is the point-to-point design where the policy earns the realized return on the index over a certain period of time at a prescribed participation rate, but with a minimal guarantee. The most popular EIA is the ratchet EIA, which represents 85% of the current market (see annuityadvantage.com). The return of ratchet EIAs is reset annually, and in each year it is the maximum of the prescribed portion of the index return and the minimal guarantee.

Because of their popularity, EIAs have received considerable attention in the actuarial literature. In a Black-Scholes framework, Tiong (2000) derived explicit prices for some popular EIAs using the Esscher transform. Lee(2003) extended Tiong’s method to some other path-dependent EIAs. Lin and Tan (2003) and Kijima and Wong (2007) argued that the effects of stochastic interest rates are crucial in pricing EIAs due to their long-term maturity. Assuming stochastic interest and mortality rates,Qian et al.(2010) priced

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the ratchet EIA analytically. For most designs it is possible to price EIAs analytically in the Black-Scholes framework because of the Markovian property of the return. Little variations on the financial model have been considered except forCheung and Yang(2005) and Lin et al. (2009) in which a Markov regime-switching model was applied to evaluate an optimal surrender strategy and price.

In our paper, we use Heston’s stochastic volatility model which is a popular method to generalize the Black-Scholes model. Since there is a closed-form for the price of European call option in Heston’s model,MacKay (2011) priced the point-to-point EIA analytically by transforming its payoff to a European call’s payoff. However, there is no closed-formula for the price of a ratchet EIA due to its complexity. Lin and Tan (2003) and Lin et al.

(2009) valued ratchet EIAs by simulation, which makes it difficult to evaluate the Greeks. In this paper, we introduce two approaches to evaluate the price and Greeks of the ratchet EIA.

The structure of this thesis is as follows: In Chapter 1, we describe the Heston Model. First, we give a review of financial frameworks. Since Heston’s assumption of a constant interest rate does not hold for long-term investments, we generalize Heston’s model to a case of deterministic interest and give a semi-closed expression for the price of Euro-pean call options. In the end, using a global optimization algorithm, named differential evolution, we calibrate Heston’s model using observed European call option prices.

Chapters 2 and 3 introduce two methods to evaluate the prices of ratchet EIAs. The first method is called the joint transition probability density function (JTPDF) approach.

Lipton (2001) and Lamoureux and Paseka (2009) derived different explicit formulas for the JTPDF of the Heston process. We generalize this formula to the case of deterministic interest. Though the formula is analytic, it is hard to calculate the oscillatory integral in it. Following Ballestra et al.(2007), we solve the oscillatory integral by the Filon-type quadrature. Given the JTPDF, pricing ratchet EIAs boils down to a question of solving multiple integrals. Again, followingBallestra et al.(2007) the multiple integrals are solved by the so-called importance sampling technique. However, our samples are generated by

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Quasi-Monte Carlo methods.

We call the other method the conditional expectation (CE) approach. Conditioning on the volatility path, we first price the ratchet EIA in a Black-Scholes framework. Since there are explicit formulas for the prices of ratchet EIAs in the Black-Schloles model, we can evaluate its price in Heston framework by simulating the stochastic volatility. The idea of taking conditional expectation can be dated back to Hull and White (1987) in which stochastic volatility was first introduced in Finance. Broadie and Kaya (2004) adopted the same idea to evaluate the Greeks for European call and Asian options. In terms of the CE method, the key point is how to simulate the integrated variance. Broadie and Kaya

(2006) and Glasserman and Kim (2008) introduced exact simulation schemes, but ac-cording to Tse and Wan (2010) and B´egin et al. (2012) both exact schemes are time-consuming. In our paper, we approximate the integrated variance by a summation of gamma distributions, which is faster and with acceptable errors.

Chapter 4 discusses the equity-indexed annuities, especially the ratchet EIA. We ap-plied the JTPDF and CE methods to evaluate ratchet guarantees. The Greeks are also derived using the JTPDF or CE methods. Numerical results are presented in the last chapter. Some sensitivity tests are also conducted in Chapter 5.

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1

Economic Model

1.1

From Black-Scholes to Stochastic Volatility

Since the introduction of Black-Scholes model in 1973, there have been a lot of empirical examples showing that the assumption of constant volatility does not correctly describe stock returns. Firstly, Blattberg and Gonedes (1974) show that the log-return of the s-tock does not follow the normal distribution but a leptokurtic density, which has heavier tails and a higher peak. However, this phenomenon can be explained by time dependent stochastic volatilities and people realized that the volatility also has a mean-reverting property. Furthermore,Beckers (1980) presents statistical evidence of a negative relation-ship between the level of stock price and its volatility. Nandi (1998) evaluate the corre-lation between the stochastic return and the volatility and its importance. The volatility smile obtained from the implied volatility casts more doubts on the Black-Scholes model. In order to describe these behaviors such as a leptokurtic density, mean-reverting property, negative correlation and volatility smile, a proper model is needed to describe the variability of the volatility. Cox (1975) develops a constant elasticity of variance (CEV) diffusion model. It assumes that the volatility is a decreasing function of the stock price. Derman and Kani (1994), Dupire (1994) andRubinstein (1994) suggest to use the so-called local volatility model, i.e. the volatility should be a deterministic function of the stock price and time. They also develop appropriate binomial or trinomial option pricing procedures. However, Dumas et al. (1998) prove that although the deterministic volatility (DV) model beats constant volatility models, the DV model’s predictions get worse with the complexity of the assumptions on the volatility and the hedge ratios are not as reliable as those in Black-Scholes model.

Stochastic volatility models describe the volatility using a diffusion process. Denote by Vt the stochastic volatility, then the general process is given by

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where p(St, Vt, t) and q(St, Vt, t) are functions of St, Vt and t, and W(t) is a standard

Brownian motion. Different p(St, Vt, t) and q(St, Vt, t) lead to different models (here, we

do not consider cases with jumps). Research works on stochastic volatility can be found in Johnson and Shanno (1987), Wiggins (1987), Scott (1987), Hull and White (1987),

Stein and Stein (1991), Heston (1993), Sch¨obel and Zhu (1999), Lewis (2000), and Zhu

(2010). Table 1, which is from Zhu (2010), gives an overview of some representative stochastic volatility models. In Table 1 Vt means the stochastic volatility and vt stands

Table 1: Stochastic volatility models

Johnsom and Shanno (1987) (1): dVt=κVtdt+σVtdW(t)

Wiggins (1987) (2): dlnVt=κ(θ−lnVt)dt+σdW(t)

Hull and White (1987) (3): dVt2 =κVt2dt+σVt2dW(t)

Hull and White (1987) (4): dV2

t =κVt2(θ−Vt)dt+σVt2dW(t)

Stein and Stein (1991) (5): dVt=κ(θ−Vt)dt+σdW(t)

Heston (1993) (6): dvt =κ(θ−vt)dt+σ√vtdW(t) Lewis (2000) (7): dvt =κ(θ−vt2)dt+σv 3/2 t dW(t) Zhu (2010) (8): dv(t) = κθ− √vt−λvt dt+σ√vtdW(t)

for the stochastic variance so thatVt=√vt. Among the models given in Table1, only (5)

(6), (8) have analytic option prices in the case of non-zero correlation between the stock returns and volatility. Models (1) and (3) can not produce the mean-reverting property.

Zhu(2010) shows that models (1), (2), (3), (4), (7) are not stationary processes, so they violate the feature of stationarity of volatility or variance. Hence models (5), (6), (8) are worth studying in details.

In fact, each stochastic volatility model in Table 1has a corresponding discrete

mod-el. Discretizing them leads to autoregressive random variance models. The generalized autoregressive conditional heteroscedasticity (GARCH) models also play important roles in studying volatility, but since we only focus on continuous models this kind of approach

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will not be considered here.

1.2

Heston’s Stochastic Volatility

In this paper, we assume that the asset price St follows the Heston model which, under

the real probability measure P, is described as follows:

dSt =μStdt+√vtStdWs(t), dvt =κ(θ−vt)dt+σ √ vtdWv(t), dWs(·), Wv(·)t =ρdt, (1.1)

where Ws(t) and Wv(t) are two standard Brownian motions with negative correlation ρ, μis the drift andv0 andS0 are known. In general, the initial asset priceS0 is observable from the market and v0 can be calibrated.

In this model, the asset price St still follows a geometric Brownian motion, but with

volatility √vt. Instead of modeling the stochastic volatility directly, Heston described

the variance vt by the mean-reverting square root process with long-run mean θ, rate

of reversion κ, volatility of volatility σ. The mean-reversion is a desired property for stochastic volatility or variance and is well documented by many empirical studies. The process for vt is the square root process, which is called the Cox-Ingersoll-Ross (CIR)

process. Cox et al. (1985) apply this process to model interest rates. It is also called the Feller process because of William Feller’s early work on this process. The CIR process has the following properties.

Proposition 1.1 For the CIR process

dvt=κ(θ−vt)dt+σ

vtdWv(t).

1. The conditional probability density function of vt+τ given its current valuevt can be

expressed as p(vt+τ|vt) =ce−ξ−ν ν ξ q/2 Iq 2ξν , (vt, vt+τ ≥0). (1.2)

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where c = 2κ σ2(1e−κτ), ξ = c·vt·e−κτ, ν = c·vt+τ, q = 2κθ σ2 −1.

and Iq[·] is the modified Bessel function1 of the first kind of order q.

2. The conditional expectation of vt+τ givenvt is

E[vt+τ|vt] =θ+ (vt−θ)e−κτ. (1.3)

Proof.

See Cox et al. (1985).

For a given t, the random variable 2cvt follows a noncentral chi-square with 2q + 2

degrees of freedom and parameter of noncentrality 2ξ. When the Feller condition (2κθ > σ2) is satisfied, the process v

t is strictly positive.

Heston’s stochastic volatility model is one of the most popular stochastic models for equities because of the following reasons. Firstly, a suitable set of the Heston parameters {κ, θ, ρ, σ} can produce a leptokurtic distribution (high peak and heavy tails) of asset returns and the negative correlation between the asset price and volatility. Examples are available in Moodley (2005). Secondly, the volatility has the mean-reverting property. Thirdly, it captures the volatility smile. Finally, a closed-formula for the European call option is available so that it is possible to calibrate the Heston model.

1.3

Deterministic Interest Rate

Heston (1993) assumes that the interest rate is a constant. It is acceptable for

short-1The definition of the modified Bessel function of first kind can be found inBowman(1958) and it is

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term investment products since their maturities usually are less than 2 years. In such a relatively low interest rate market, a constant interest rate could give an acceptable approximation. However, for long-term investment products whose maturities range from 5 to 15 years, the risk-free interest rate is more volatile. Table2in Section 1.5 illustrates

the observed yield rates of T-bills on August 9th, 2011, which are regarded as risk-free investment products. The maturities of the T-bills in Table 2 range from 1 month to

30 years, and the table shows that after 2 years the yield rate varies a lot. Hence, it is unreasonable to assume a constant risk-free interest rate when we are dealing with long-term investment products.

The constant assumption could be overcome by assuming deterministic or stochastic interest rate models. A deterministic risk-free rate is described by a function of time t, while a stochastic process is used to represent a stochastic risk-free rate. Though a deter-ministic interest rate model is not as accurate as a stochastic model, some deterdeter-ministic models still fit the data correctly in the long run. Lin and Tan (2003) apply stochastic interest rates to equity-indexed annuities (EIAs), and show how the interest rate affects participation rates. However, their results show that randomness just introduce little impact on the participation rates. Moreover, the problem gets more complex when s-tochastic interest rate models are introduced in ss-tochastic volatility models. Hence, we use deterministic interest in this paper.

Instead of modeling the risk-free interest rate rt directly, we turn to the yield rates of

T-billsyt. Present value factor can be obtained from yield rates using e−0trsds =e−tyt.

1.4

Non-Arbitrage Assumption and Heston Framework

Generally speaking, the non-arbitrage assumption says that it is impossible to make money out of nothing. This concept is important in mathematical finance.

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Black and Scholes (1973) derive a PDE for option pricing under the non-arbitrage assumption and solve the PDE, leading to a closed-form equation for the price of European call options. Harrison and Pliska (1981) prove that under the non-arbitrage assumption, there is a risk neutral measure Q under which the asset price earns a risk-free interest rate and introduced the popular risk-neutral pricing formula. Under the non-arbitrage assumption, Heston (1993) also derives a PDE for prices of European call options, and also gives the asset prices dynamic underQ.

Adopting the non-arbitrage assumption, the Heston model with deterministic interest underQ is given by:

Proposition 1.2 Heston’s process under the risk-neutral measure Q is given by

dSt=rtStdt+ √ vtStdWsQ(t), dvt=κ∗(θ∗−vt)dt+σ √ vtdWvQ(t), dWsQ(·), WvQ(·)t=ρdt, (1.4)

whereWsQ(t)and WvQ(t)are standard Brownian motions underQ; rt is the risk-free rate. κ∗ = κ+λ0 and θ∗ = κ+κθλ

0. Here, {κ, θ, σ, ρ} are the same as the parameters in (1.1).

Finally,λ0, a constant, is related to the market price of the volatility risk.

Proof. For the value of an option Π, a PDE can be derived according to (A.11) in Appendix A. That is 1 2vtS 2 t ∂2Π ∂S2 t +1 2σ 2v t ∂2Π ∂v2 t +ρσStvt ∂2Π ∂vt∂St +∂Π ∂t −rtΠ +rtSt ∂Π ∂St + (κ(θ−vt)−λ(s, v, t)σ √ vt) ∂Π ∂vt = 0. (1.5)

In Heston’s model,λ(St, vt, t) is assumed to be λ0√vt

σ , that is the market price of volatility

risk.

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rtSt and κ(θ−vt)−λ0vt respectively. Hence, (1.1) can be rewritten as follows, dSt St =rtdt+ √ vt μ−rt √ vt dt+dWs(t) , dvt= (κ(θ−vt)−λ0vt)dt+σ √ vt λ0√vt σ dt+dWv(t) , dWs(·), Wv(·)t=ρdt,

Qis just the measure under which

μ−rt √ vt dt+dWs(t) = dWsQ(t), λ√vt σ dt+dWv(t) = dW Q v (t).

One term which calls for attention is the market price of volatility risk. Though it does not appear in (1.1), the PDE in (1.5) shows that it has an impact on the option

prices. However, the market price of volatility risk is hard to estimate since the volatility is not traded in the financial market. In Heston (1993) it is assumed to be λ0√vt

σ . This

assumption at least fits common sense, as the market price of volatility should be higher when the volatility is large and lower when vt is small. Moreover, under this assumption

the asset prices keep the same dynamic under both measuresP and Q.

1.5

European Call Price

In order to price and hedge derivatives in the Heston framework, we need to identify the parameters in (1.4), these are {κ∗, θ∗, σ, ρ, v0}. The parameters are calibrated by comparing the observed European call prices and the prices obtained using the Heston model. The closed-form expression for European call prices is derived in Heston’s original paper, but as we pointed out in the previous section, the assumption of constant interest does not hold for pricing EIAs. In this section, following Heston (1993) and by studying the characteristic function of the log-return, we give an analytical formula for the price of European calls under Heston model with deterministic interest.

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1.5.1 Characteristic Functions

The payoff of European calls is max{ST −K,0}, where ST is the asset price at maturity T and K is the strike price. It means that if the asset price at maturity ST is larger than

the strikeK then the payoff of one unit of European call is ST−K. Otherwise, the payoff

is zero. In order to price European call options, we use the risk-neutral valuation formula which was introduced in Harrison and Pliska (1981) as follows:

Proposition 1.3 Denote by Y(T)the valuation of an attainable2 claim exercised at time

T, then its price at time t is given by

Y(t) =B(t)EQ Y(T) B(T) Ft , (1.6) where B(t) =e0trwdw

and Ft is the filtration (or the information) up to time t.

Proof. See Harrison and Pliska (1981).

Remark 1.4 B(t)is also called the numeraire3underQsuch that BY((tt)) is aQ-martingale.

According to Proposition1.3, the price of a European call option, ΠC, is given by

ΠC(t, T, St, vt, K) = EQ e− T t rwdwmax{ST −K,0}Ft = e−tTrwdwEQ[STI{ST > K}|Ft] −Ke− T t rwdwQ{ST > K|Ft}, (1.7)

where Q{ST > K|Ft} is the probability of the event {ST > K|Ft} under Q and I{·}

stands for the indicator function.

In order to evaluate EQ[STI{ST > K}|Ft] we change Q to another measure Qs by

the technique of change of numeraire. Before doing that, it is necessary to introduce two lemmas.

2A contingent claim attainable if it can be replicated by a self-financing portfolio.

3A numeraire is a price processX(t), almost surely strictly possitive for each t[0, T]. For details of

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Lemma 1.5 Assume that B(t), Q, Ft are as in Proposition 1.3 and let X(t) be a

non-dividend paying numeraire such that X(t)/B(t) is a Q-martingale. Then there exists a probability measure QX defined by its Radon-Nikodym derivative with respect to Q, that

is

dQX

dQ |FT =

B(0)/B(T)

X(0)/X(T),

such that the basic security prices discounted with respect to X are QX martingales.

Proof. See Geman et al.(1995).

Lemma 1.6 (Bayes Formula) Assume that P˜ is absolutely continuous4 with respect to P

andZis its Radon-Nikodym derivative with respect toP. If Y is bounded (orP˜-integrable) and FT measurable, then

E˜P[Y|Ft] =

1

Z(t)E

P[YZ(T)|F

t], a.s. for t≤T.

Proof. See Exercises 5.1 in Bingham and Kiesel (2004).

Proposition 1.7 Given the Heston model (1.4), let B(t) be the numeraire under Q. In other words, B(t) stands for the money market account e0trwdw. Then

1. St/B(t) is a Q-martingale.

2. There exist a measure QS such that the basic security prices discounted with respect

to St are QS martingales.

3. Heston’s model under the measure QS is given by dSt = (rt+vt)Stdt+ √ vtStdWsQS(t), dvt = [κ∗(θ∗−vt) +ρσvt]dt+σ √ vtdWvQS(t), dWQS s (·), WvQS(·)=ρdt, (1.8) where WQS

s (t) and WvQS(t) are two standard Brownian motions underQS.

4Measureμ is absolutely continuous with respect toν is there exists a functiong such that μ(A) =

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Proof.

1. Apply Itˆo’s formula, we have

d St B(t) = dSt B(t) +StdB(t) +dS·, 1 B(·)t = 1 B(t) rtStdt+ √ vtStdWsQ(t) +St − 1 B(t)2 B(t)rtdt = 1 B(t) rtStdt+ √ vtStdWsQ(t)−rtSt = 1 B(t) √ vtStdWsQ(t).

2. This is true because of Lemma 1.5 and the previous result.

3. Let Yt = lnSt, applying Itˆo’s formula to Yt leads to

dYt = ∂Yt ∂St +1 2 ∂2Y t ∂S2 t dS·, S·t+ ∂Yt ∂t dt = 1 St dSt+ 1 2 − 1 S2 t St2vtdt = rt− 1 2vt dt+√vtdWsQ(t). Hence, lnSt S0 =Yt−Y0 = t 0 rs− 1 2vsds+ t 0 √ vsdWsQ(s)ds.

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Then, according to Lemma1.5, dQS dQ = B(0)/B(t) S0/St = 1 e0trsds exp t 0 rs− 1 2vsds+ t 0 √ vsdWsQ(s)ds = exp t 0 √ vsdWsQ(s)ds− t 0 1 2vsds . (1.9) By Girsanov’s theorem dWQS s (t) =dWsQ(t)− √ vtdt. (1.10) Since dWQS v (t) can be written as ρdWsQS(t) + 1−ρ2dZQS(t), where ZQS(t) is a standard Brownian Motion under QS which is independent ofWsQS(t), then

dWQS v (t) = ρdWsQS(t) + 1−ρ2dZQS(t) = ρdWsQ(t) +1−ρ2dZQS(t)−√v tdt = ρdWsQ(t) + 1−ρ2dZQ(t)−√v tdt (1.11) = ρdWvQ(t)− √ vtdt. (1.12)

Equation (1.11) holds since while we change the measure by (1.9), ZQ(t) is still a

standard Brownian motion under the new measure, that is ZQS(t) =ZQ(t). Hence, (1.8) is true because of (1.10) and (1.12).

Remark 1.8 Wong and Heyde (2006) prove that St

B(t) is a true martingale only if κ∗ ≥ σρ. In Part1of Proposition1.7, the conditionκ∗ ≥σρis not necessary since by definition

κ∗ and σ must be larger or equal to 0, and in (1.1) we assume that ρ <0.

By the definition ofQS, both QS andQ are absolutely continuous to each other. Denote

the Radon-Nikodym derivative of Qwith respect to QS by Zt, s.t.

Zt = dQ dQS Ft= S0/St B(0)/B(t),

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by Lemma 1.6 we have EQ[STI{ST > K}|Ft] = 1 Zt EQS[Z TSTI{ST > K}|Ft]. Hence, e−tTr(w)dwEQ[STI{ST > K}|Ft] = B(t) B(T)E Q[S TI{ST > K}|Ft] = B(t) B(T) 1 Zt EQS[Z TSTI{ST > K}|Ft] = EQS B(t) B(T) ZT Zt STI{ST > K}|Ft = EQS[S tI{ST > K}| Ft] = StQS{ST > K|Ft}.

where QS{ST > K|Ft} stands for the probability of the event {ST > K|Ft} under

mea-sure QS. Then the call price (1.7) can be written as

ΠC(t, T, St, vt, K) = StQS{ST > K|Ft} −Ke− T t rwdwQ{ST > K|Ft} = StP1−Ke− T t rwdwP2 extP 1−Ke− T t rwdwP2. (1.13) whereP1 QS{ST > K|Ft}, P2 Q{ST > K|Ft} and xt= lnSt.

We turn to the log-pricextin order to simplify the calculations. Applying Itˆo’s formula

toxt, we obtain the SDE for the log-price as follows, dxt= rt− 1 2vt dt+√vtdWsQ(t), dvt=κ∗(θ∗−vt)dt+σ √ vtdWvQ(t), dWsQ(·), WvQ(·)t=ρdt. (1.14)

Hence, to obtain a formula for Π(t, T, St, vt, K) we just need to seek forP1andP2.

Where-as,Rollin et al.(2010) reported that the the probability density function of the log-return or log-price is still not well known, and maybe do not have a closed-form. However, the characteristic function of the log-price has nice properties. Hence the valuation of options

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via characteristic function (CF) is popular: in fact inHeston (1993), the original closed-form for call price is derived using the CF;Bakshi and Madan(2000) shows that CF plays a significant role in pricing options and simplify the problem; Dr˘agulescu and Yakovenko

(2002) derived the PDF of the “log-return” in Heston’s framework via CF, assuming the initial stochastic variance follows a stationary distribution. Zhu (2010) gave a thorough description of applying the CF to Heston’s model. The following results explain how to derive the call price via the characteristic function of the log-price.

Proposition 1.9 Given the characteristic function φ(s) of a random variable X, the CDF of X, FX(x), is given by FX(x) = 1 2 − 1 2π −∞ e−isxφ(s) is ds. (1.15)

Proof. See Gil-Pelaez (1951).

Corollary 1.10 Assume that φj(s) is the characteristic function corresponding to Pj,

for j = 1,2 in (1.13), then we have the following relationship:

Pj = 1 2+ 1 π 0 Re e −islnKφ j(s) is ds, (1.16)

where Re[z] is the real part of z.

Proof. According to Proposition1.9, for j = 1,2,

Pj = 1−F(lnK) = 1 2+ 1 2π −∞ e−islnKφ j(s) is ds = 1 2+ 1 2π − 0 +∞ e−i(−s˜) lnKφj(−s˜) −i˜s ds˜+ 0 e−islnKφj(s) is ds = 1 2+ 1 2π +∞ 0 ei˜slnKφj(˜s) −i˜s d˜s+ 0 e−islnKφj(s) is ds (1.17) = 1 2+ 1 π 0 Re φj(s) e−islnK is ds, (1.18)

where a+ib=a−ib, a, b ∈R, Im[z] and Re[z] denote the imaginary and real part of

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Equation (1.17) is true since for a characteristic function φ(−s) = φ(s) and (1.18) holds since the probability density Pj must be real, so that the integrals w.r.t the

imagi-nary part can be eliminated.

Note that Corollary 1.10is model free, so that the problem of pricing a European call option can be converted to a problem of identifying the characteristic functions of the asset price (or log-price) underQ and QS. In Heston’s model, φj, j = 1,2, are obtained

through the Heston PDE.

1.5.2 Heston PDE

In order to derive the Heston PDE, we need to introduce the following theorem.

Theorem 1.11 (Feynman-Kac Theorem)

Suppose that

1. xt follows the stochastic process in n dimensions

dxt=μμμ(xt, t) +σσσ(xt, t)dWWWQ(t), (1.19)

where xt andμμμ(xt, t) are n-dimensional column vectors,σσσ(xt, t) is an×m matrix

andWWWQt is a m-dimensional Q−Brownian motion. That is

d ⎛ ⎜ ⎜ ⎜ ⎝ x1(t) .. . xn(t) ⎞ ⎟ ⎟ ⎟ ⎠= ⎛ ⎜ ⎜ ⎜ ⎝ μ1(xt, t) .. . μn(xt, t)) ⎞ ⎟ ⎟ ⎟ ⎠dt+ ⎛ ⎜ ⎜ ⎜ ⎝ σ11(xt, t) · · · σ1m(xt, t) .. . . .. ... σn1(xt, t) · · · σnm(xt, t) ⎞ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎝ dW1Q(t) .. . dWmQ(t) ⎞ ⎟ ⎟ ⎟ ⎠,

where μi(xt, t) and σij(xt, t) are functions from Rn+1 to R.

2. By definition, the generator of the process xt is

A = n i=1 μi ∂ ∂xi +1 2 n i=1 n j=1 σ σσσσσT i,j ∂2 ∂xi∂xj , (1.20)

where for convenience μi =μi(xt, t), σσσ =σσσ(xt, t) and

σ σ

σσσσTij is the element (i, j)

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Now let f ∈C2 0(R

n), qC(Rn) and is lower bounded, we have

1. Put Y(t,x) =E exp − t 0 q(xs) ds f(xt) x , (1.21) then ∂Y ∂t =AY −qY; t >0, xR n , (1.22) Y (0,x) =f(x) ; xRn. (1.23)

2. Moreover, ifw(t,x)∈C1,2(R×Rn)is bounded onK×Rnfor each compactK ⊂R and w solves (1.22), and (1.23), then w(t,x) =Y (t,x), given by (1.21).

Proof. See Oksendal (2002).

The Heston PDE can be derived directly from Theorem 1.11. Recall that in the Heston framework the asset price dynamics under the risk-neutral measure are described by (1.14). The process for xt = (xt, vt) can be written in terms of two independent

Brownian motionsZ1 and Z2 as

d ⎛ ⎝xt vt ⎞ ⎠= ⎛ ⎝ rt−12vt κ∗(θ∗ −vt) ⎞ ⎠dt+ ⎛ ⎝ √vt 0 σρ√vt σ vt(1−ρ2) ⎞ ⎠ ⎛ ⎝dZ1(t) dZ2(t) ⎞ ⎠. (1.24)

To apply the Feymann-Kac Theorem, we need the generator given in (1.20). Since

σ σσσσσT = ⎛ ⎝ √vt 0 σρ√vt σ vt(1−ρ2) ⎞ ⎠ ⎛ ⎝ √vt 0 σρ√vt σ vt(1−ρ2) ⎞ ⎠ T = ⎛ ⎝ vt ρσvt ρσvt σ2vt ⎞ ⎠,

the generator in (1.20) becomes

A = rt− 1 2vt ∂ ∂xt +κ∗(θ∗ −vt) ∂ ∂vt +1 2 vt ∂2 ∂x2 t + 2σρvt ∂2 ∂xt∂vt +σ2vt ∂2 ∂v2 t . (1.25)

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Then (1.22) can be written as ∂Y ∂t + rt− 1 2vt ∂Y ∂xt +κ∗(θ∗−vt) ∂Y ∂vt +1 2vt ∂2Y ∂x2 t +σρvt ∂2Y ∂xt∂vt +1 2σ 2v t ∂2Y ∂v2 t −rtY = 0. (1.26) Equation (1.26) is called the Heston PDE which is used to solve forφj and j = 1,2.

1.5.3 Solving the Characteristic Functions

Note that the definition ofY in (1.26) is given by (1.21), we can replaceY by ΠC in (1.26)

and get the following PDE by setting τ =T −t, −∂ΠC ∂τ + 1 2vt ∂2Π C ∂x2t + rT−τ − 1 2vt Π C ∂xt +ρσvt ∂2Π C ∂vt∂xt +1 2σ 2v t ∂2ΠC ∂v2 t −rT−τΠC + [κ∗(θ∗−vt)] ∂ΠC ∂vt = 0. (1.27) Inserting (1.13) into (1.27) leads to

ext −∂P1 ∂τ + rT−τ + 1 2vt ∂P1 ∂xt +1 2vt ∂2P 1 ∂x2 t +ρσvt ∂2P 1 ∂vt∂xt + [ρσvt+κ∗(θ∗−vt)] ∂P1 ∂vt + 1 2σ 2v t ∂2P1 ∂v2 −Ke−TT−τr(w)dw −∂P2 ∂τ + 1 2vt ∂2P2 ∂x2 t + rT−τ − 1 2 ∂P2 ∂xt +ρσvt ∂2P1 ∂xt∂vt + κ∗(θ∗−vt) ∂P2 ∂vt +1 2σ 2v t ∂2P 2 ∂v2 t = 0. (1.28)

Since (1.28) must be true for all strike values, the term multiplying K and the term

independent of K should each be zero. Hence, (1.28) can be rewritten into two PDEs,

that are −∂P1 ∂τ + 1 2vt+rT−τ ∂P1 ∂xt + 1 2vt ∂2P 1 ∂x2 t +ρσvt ∂2P 1 ∂vt∂xt + [ρσvt+κ∗(θ∗−vt)] ∂P1 ∂vt +1 2σ 2v t ∂2P1 ∂v2 t = 0, (1.29) and − ∂P2 ∂τ +ρσvt ∂2P2 ∂vt∂xt +1 2 ∂2P2 ∂x2 t + 1 2vtσ 2∂2P2 ∂v2 t + rT−τ− 1 2vt ∂P2 ∂xt + [κ∗(θ∗−vt)] ∂P2 ∂vt = 0. (1.30)

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We can rewrite (1.29) and (1.30) by −∂Pj ∂τ +ρσvt ∂2P j ∂xt∂vt +1 2vt ∂2P j ∂x2 t + 1 2vtσ 2∂2Pj ∂v2 t + (rT−τ+ujvt) ∂Pj ∂xt + (a−bjvt) ∂Pj ∂vt = 0, (1.31) where u1 = 1 2, u2 =− 1 2, a=κ ∗θ, b 1 =κ∗−ρσ, b2 =κ∗.

In order to obtain PDEs ofφj, for j = 1,2, we need to introduce Kolmogorov’s backward

equation given in the following theorem.

Theorem 1.12 (Kolmogorov’s backward equation)

Let f ∈C2 0(R

n) and x

t be and Itˆo diffusion in Rn with generator A.

1. Define

u(t,x) =Ef(xt)x

. (1.32)

Thenu(t,·)∈ DA for each t and

∂u

∂t =Au, t >0, xR n

, (1.33)

u(0,x) =f(x) ; xRn, (1.34)

where the right hand side is to be interpreted as A applied to the function x

u(t,x).

2. Moreover, if w(t,x)∈C1,2(R × Rn)is a bounded function satisfying (1.33), (1.34)

then w(t,x) = u(t,x), given by (1.32).

Proof. See Oksendal (2002).

According to Theorem 1.12, the characteristic functions φj must satisfy the PDEs of Pj but with different boundary conditions, which are

φj

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In fact, by definition whenτ = 0

φ1(s) =EQSeisx0 =eisx0 , φ2(s) =EQeisx0 =eisx0

.

FollowingHeston (1993) we assume the characteristic function is the form of:

φj(s) = exp{Cj(τ, s) +Dj(τ, s)vt+isxt}. (1.35)

Hence, Dj(0, s) = 0 and Cj(0, s) = 0. Inserting (1.35) into (1.31) leads to

− ∂Cj ∂τ + ∂Dj ∂τ vt +ρσvtisDj− 1 2vts 2+1 2vtσ 2D2 j + (rT−τ +ujvt)is+ (a−bjvt)Dj = 0.

Grouping the terms including vt, we have vt −∂Dj ∂τ +ρσisDj − 1 2s 2+ 1 2σ 2D2 j +ujis−bjDj − ∂Cj ∂τ +rT−τis+aDj = 0. (1.36)

Note that (1.36) must be true for all values of vt, so that in (1.36) the term with vt and

the term free fromvt are both supposed to be zero. Hence, (1.36) can be rewritten using

the following two equations:

∂Dj ∂τ = ρσisDj − 1 2s 2+1 2σ 2D2 j +ujis−bjDj (1.37) ∂Cj ∂τ = rT−τis+aDj. (1.38)

Since D(0, s) = 0, (1.37) is solvable according to Appendix C. Setting

Lj =ujis− 1 2s 2, Q j =ρσis−bj and R= 1 2σ 2, (1.37) can be written as ∂Dj ∂τ =Lj +QjDj +RD 2 j.

According to Appendix C, denote by

B0 = −Qj+ Q2 j −4RLj 2R , A = 2RB0+Qj,

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Dj = B0(A−RB0)e−At (A−RB0)B0 (A−RB0)e−At+RB 0 = A−RB0 R e−Aτ 1 A−RB0 RB0 e− Aτ + 1 = ρσis−bj −dj σ2 edjτ 1 ρσis−bj−dj −ρσis+bj−dje djτ + 1 = bj−ρσis+dj σ2 1−edjτ 1−gjedjτ , (1.39) where dj = − (ρσis−bj)2−σ2(2ujis−s2), gj = bj −ρσis+dj bj−ρσis−dj .

Given Dj, the solution for Cj with boundary conditions Cj(0, s) = 0 is given by

integrating (1.38), Cj = τ 0 r(T −τ)is dτ +a bj −ρσis+dj σ2 τ 0 1−edjτ 1−gjedjτ dτ = τ 0 r(T −τ)is dτ + a σ2 (bj −ρσis+dj)τ−2ln 1−gjedjτ 1−gj = is T t r(w)dw+ a σ2 (bj−ρσis+dj)τ −2ln 1−gjedjτ 1−gj . (1.40)

ObtainingCj and Dj, (1.35),(1.39) and (1.40) give us the formula forφj,j = 1,2. Hence,

the formula of call price could be obtained by combining (1.35), (1.17) and (1.13).

Though a closed (or semi-closed) formula for the European call price is derived, there are still some numerical problems which make it unpractical. Albrecher et al. (2007) pointed out that due to the discontinuity in the branch cut of the complex logarithm in

C1, φ1(s) is unstable for the long term maturity or for certain parameters, while φ2(s) does not suffer this problem. This numerical problem was solved by Bakshi and Madan

(2000) in which the relationship between φ1(s) and φ2(s) was found as follows:

φ1(s) = φ2(s−i)

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Sinceφ1 in our paper suffers the same problem as Heston’s original formula, we use φ2(s) and the relationship between the two characteristic functions given in (1.41) to evaluate

P1 and P2.

Finally, we give the European call price under Heston model with deterministic interest as follows: Under Heston’s model with deterministic interest, the price for a European call option with strikeK and maturity T at timet(≤T) given current asset priceSt and

current stochastic variance vt can be expressed as

ΠC(t, T, St, vt, K) =extP1−Ke− T t rwdwP 2, (1.42) where xt = lnSt, P1 = 1 2 + 1 π 0 Re e −islnKφ 2(s−i) isφ2(−i) ds, P2 = 1 2 + 1 π 0 Re e −islnKφ 2(s) is ds, φ2(s) = exp{C(τ, s) +D(τ, s)vt+ixts}, C(τ, s) = is T t r(w)dw+κ ∗θ∗ σ2 (κ ∗ρσis+d)τ 2 ln1−gedτ 1−g , D(τ, s) = κ ∗ρσis+d σ2 1−edτ 1−gedτ , g = κ ∗ ρσis+d κ∗−ρσis−d, d = − (ρσis−κ∗)2+σ2(is+s2), τ = T −t.

Please note that “d” in our paper is the opposite to the “d” in Heston’s original paper. In fact, there could be two values for “d” because it is the square root of a complex number and they lead to the same result. This is because of the uniqueness of the solution to the Riccati function (1.37). (See details in Appendix C). However, compared with the call

prices obtained by simulation and the numerical methods that we introduced in Chapter 3, Heston’s choice generates a little error in practice while the other value of “d” leads

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to consistent results. This is due to the branch cut of the square root function of a complex number. Another explanation is that when d is positive the computation error in evaluating edτ is larger than e−dτ.

1.6

Calibrating the Heston Model

The parameters of the asset price dynamic are calibrated using observed European call prices. Since most of our formulas are derived under the risk neutral measure, we pay more attention to the calibration of the Heston parameters G2 ={κ∗, θ∗, σρ, v0}.

In order to calibrate the Heston model with deterministic interest, we first identify and calibrate the model for yield rates. In this thesis, we assume that

yt =β1+β2 1−e−t/λ1 t/λ1 +β3 1−e−t/λ1 t/λ1 −e −t/λ1 +β4 1−e−t/λ2 t/λ2 −e −t/λ2 , (1.43) where t is the time to maturity. β1, β2, β3, β4, λ1 and λ2 are parameters to be estimated. This model, which is called Nelson-Siegel-Svensson (NSS) model, is introduced inSvensson

(1994). Although we specify a certain formula for the deterministic yield rate, our financial model does not limited to the particular one.

We calibrate G1 = {β1, β2, β3, β4, λ1, λ2} according to the observed rates from the market. Table2illustrates the yield rates of the U.S. Treasury bills (T-bills)5 which were observed on August 9th, 2011 and the maturities of corresponding T-bills range from 1 month to 30 years. Following Gilli et al. (2010), we use DE method to calibrated G1

according to the yield rates given in Table 2 (brief introduction to the DE algorithm is given in AppendixB). The objective function is

G1 = arg min Ω1 i yiObs−yiN SS2,

where Ω1 is the space of parameters, yObs

i stands for the observed yield rates given by the

U.S. Department of Treasury and yiN SS is for the yield rates obtained from (1.43). The

calibration results are given in Table3.

5Data was obtained from

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Table 2: Yield curve in percentage

Time 1 Month 3 Month 6 Month 1 Year 2 Year 3 year

Yield Rate 0.02 0.03 0.06 0.11 0.19 0.33

Time 5 Year 7 Year 10 Year 20 Year 30 Year

Yield Rate 0.91 1.53 2.20 3.17 3.56

Table 3: NSS model calibration

Parameter Calibration Results

β1 4.233068 β2 −4.233048 β3 −25.918993 β4 19.522368 λ1 1.572826 λ2 1.367069

The error is estimated by the root of mean squared error which is, 1 N i yObs i −y N SS i 2 = 0.004768%.

Furthermore, Figure 1 shows the calibration results as well as the NSS model fits the

observed data well.

After calibrating the NSS model, the Heston parameters can be calibrated according to the observed data. In this paper we calibrateG2 ={κ∗, θ∗, σρ, v0}based on observed Eu-ropean call options. In other words, calibrating the Heston parameters is an optimization problem as follows, G2 = arg min Ω2 N i=1 wi ΠObsC (0, Ti, S0, v0, Ki)−ΠHestonC (0, Ti, S0, v0, Ki) 2 , (1.44)

where Ω2 is the space for G2, ΠObsC (0, Ti, S0, v0, Ki) is the observed European call option

price, ΠHeston

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0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 3.5 4 Time in Years

Yield Rate in Percent

NSS Model Observed Data

Figure 1: Calibration result for NSS model.

the number of observed prices and {wi} are the weights. Because the market prices are

given as a bid-ask spread6, it is impossible to determine the real price. Hence, the average of the bid and asked prices is used to define the observed European call prices. Following

Moodley (2005), we define the weights as | 1

bidi−aski|, where bidi is the bid price and aski is the asked price. This weight makes sense: the larger the difference between the asked and bid price, the harder it is to determine the real price, hence we are supposed to assign less weights to such prices.

In our paper, forty European call options on S&P 500 observed on 9th August, 20117 are used to calibrate Heston’s parameters. On that day the S&P 500 index closed at 1,172.53 that is used as the initial asset price S0. The maturities of the observed call options range from 2 month to 14 month and the strikes are between 800 and 1,400.

6Bid prices are the prices at which the agents sell the calls and ask prices are the prices at which the

agents buy calls.

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As we said before, the Heston model under the risk-neutral measure has five parameters that need to be estimated. Minimizing the objective function is a nonlinear programming problem. The objective function is far from being convex andMikhailov and N¨ogel(2003) pointed out that there exist many local extrema so that global optimizers should be applied here. In our paper we use the Differential Evolution (DE) method introduced by Storn and Price (1997) to minimize the objective function. The DE method is widely and successfully used when calibrating Heston model and other financial models, see

Gilli and Schumann (2010),Schoutens et al. (2004), andVollrath and Wendland (2009). The parameters estimation is given in Table 4: Note that this set of parameters

Table 4: Heston parameters under risk neutral measure

Parameter Calibration Results

κ∗ 4.1

θ∗ 0.046

σ 0.605

ρ −0.7736

v0 0.077931

satisfies the Feller condition, that is, 2κ∗θ∗ −σ2 > 0. This result is obtained using the

DE optimization algorithm with parametersnp = 50, F = 0.5, CR= 0.9, NG= 200.

The error is evaluated by the root of the weighted mean squared error, which is ! 1 N N i=1 wi ΠObsi −Π Heston i 2 = 1.4700.

Compared with the weighted average call option, that is 1 N N i=1 wiΠmodeli = 31.39

the calibration error seems acceptable. Furthermore, in Figure 2 the black stars stand for observed European call option prices and the surface is generated by (1.42) with the

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parameters given in Table 4. From Figure 2 we can see the calibrated prices fit the observed prices well.

800 900 1000 1100 1200 1300 1400 0 0.5 1 1.5 0 50 100 150 200 250 300 350 400 Maturity Strike Call Price

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2

Joint Transition PDF of Heston Model

In this section, we introduce the conditional probability density functions of the asset dynamics and volatility.

As we presented in Section 1.2, the volatility follows a noncentral chi-square distri-bution. In terms of the marginal PDF of the asset price, there is still no closed formula though a number of research works have tried to address this problem. One interesting result is given byDr˘agulescu and Yakovenko (2002). They first obtain p(ln(St/S0), vt|v0)

by inverting its characteristic function, then a formula for the marginal PDF of the asset price,p(ln (St/S0)), is derived by integratingvt and v0 out. They call their result the DY

formula and show that the DY formula fits the data of the Dow-Jones index from 1982 to 2002. Silva and Yakovenko (2003) continue to illustrate that the DY formula successfully fits the data of S&P 500 and Nasdaq indices from 1980s to 2000s. However, in integrating

v0 out they assumed it follows the stationary distribution of p(vt+τ|vt). This makes the

DY formula an approximation. Though this assumption makes sense for long-time peri-ods, it is not reasonable for short terms. Rollin et al. (2010) proved that the density of the log-return has aC∞(or smooth) density and can be written as an infinite convolution of Bessel type densities.

Though the marginal PDF of asset prices is unknown, Lipton (2001) as well as

Lamoureux and Paseka (2009) both derived closed formulas for the joint transition den-sity probability function (JTPDF) of the asset and volatility. The Heston process is described by the stochastic processes St (or xt) and vt, so that it is characterized by

the JTPDF p(xt+τ, vt+τ|xt, vt). The JTPDF is advantageous in dealing with the

path-dependent derivatives because of the Markovian property of (xt, vt). This enables us to

address the path-dependent derivatives period by period, rather than dealing with the whole path.

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2.1

The Joint Transition Probability Density Function

In this section, we generalize formulas in Lamoureux and Paseka (2009) to the case of deterministic interest.

Proposition 2.1 Given the Heston process with deterministic interest rate under Q, that is (1.4), the JTPDF of the process is given by

p(xt+τ, vt+τ|xt, vt) = 1 πe ν+1 2 κ∗τ vt+τ vt ν/2 +∞ 0 eimkh(k) dk, (2.1) where m = − xt+τ −xt− T t rwdw+ ν+ 1 2 ρστ , h(k) = ψIν[2√ϕvtvt+τ]e τ d 2 +(A2−ψ)vt−(A1+ψ)vt+τ , ν = 2κ ∗θ∗ σ2 −1, ψ = d R1(edτ −1), ϕ = ψ2edτ, A1 = 1 2R1[−R2 +d], A2 = 1 2R1[−R2 −d], R1 = σ 2 2 , R2 = ikρσ−κ∗, d = (κ∗−ikρσ)2+σ2(k2+ik),

and Iν[·] is the modified Bessel function of the first kind of order ν.

Proof. Lamoureux and Paseka (2009) give a proof when the interest rate is constant. This idea can be applied to the case of deterministic interest rates. See Ballestra et al.

(2007) on how to change the bounds of the interval.

Figure3consists of the graphs related to the JTPDF. The plot on the left is the figure of the JTPDF, the plot on the top right is the projection on the surface of JTPDF andxt,

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and the third plot at the bottom right is the projection on the surface of JTPDF andvt.

The parameters used for plotting are obtained from the calibration. Figure3 shows how

the log-price and the stochastic variance of S&P 500 would change after August 9th, 2011 based on the market of European call prices that we observed. Firstly, the support of the JTPDF is rather small compared to its domain which is [−∞,+∞]×[0,+∞]. Secondly, the log-price tends to increase with a large probability and has a heavier tail on the left side. Thirdly, the volatility tends to decrease with large probabilities.

3 3.5 4 4.5 5 5.5 0 0.1 0.2 0.3 0.4 0 10 20 30 40 50 60 Log Price volatility TPDF 3 3.5 4 4.5 5 5.5 0 10 20 30 40 50 60 Log Price XT vs TPDF TPDF 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 10 20 30 40 50 60 VT vs TPDF volatility TPDF κ* = 4.1, θ* = 0.046, σ = 0.605, ρ = 0.7736, v 0 = 0.077931, x0 = 4.6052, maturity = 1

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2.2

The Quadrature of The TPDF Integral

The integral in (2.1) is an oscillatory integral due to the term eimk. When traditional

quadrature is applied to evaluate such integrals, the result is not reliable. In fact, to evaluate an integral, for exampleI =abf(x)dx, the traditional quadrature first generates N points{xi},i= 1,2, . . . , N according to a certain rule from the interval [a, b], then the

following summation is applied to approximate the integral

I =

N

i=1

ωif(xi),

where ωi is the weight assigned to f(xi). Setting ωi = N1 leads to the Monte-Carlo

method. When f(x) is difficult to evaluate, f(x) is approximated by an interpolation

g(x) according to (xi, f(xi)), i= 1,2, . . . N. The function g(x) is selected such that it is

always easy to evaluate. Hence, the basic idea of typical quadrature is

I = b a f(x)dx ≈ N i=1 ωig(xi).

However, whenf(x) is an oscillatory function the integral of f(x) is referred to an oscil-latory integral and the points might not be representative of the behavior of the function. Figure 4 shows how these interpolation points could miss their target. In Figure 4 the

blue line is the plot for function x2cos(60x), which is a typical oscillatory function. To approximate this function we randomly generate 20 points that are marked by red s-tars. These interpolation points are fooled by the behavior ofx2cos(60x), and the typical

quadrature gives the integral of the function presented by the red line. Some Gaussian quadratures such as the adaptive Gauss-Kronrod or Lobatto quadratures works at eval-uating the oscillatory integral in (2.1). However, this is because the oscillatory factor m

in (2.1) is too small for our problem. When the oscillatory factor is large, like the func-tion in Figure4, the results obtained by a typical quadrature are not reliable. Hence, the

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−3 −2 −1 0 1 2 3 −10 −8 −6 −4 −2 0 2 4 6 8 10 Figure 4: f(x) =x2cos(60x), x∈[−3,3]

is time-consuming since the method requires a lot of interpolation points when evaluating the oscillatory integral. Moreover, the Gaussian-type quadrature gives modest accuracy. However, to evaluate an oscillatory integral there are many other numerical methods.

Olver(2008) gives a good review of the literature on oscillatory integrals. In our paper we use the Filon-type quadrature which was introduced inFilon (1928) to solve the integral

I =

b a

eimkh(k) dk. (2.2)

Though it was introduced almost one hundred years ago, it is fairly good in solving oscillatory integrals when the oscillatory factor m is given. So far there have been just limited improvements to Filon’s idea.

Filon’s method approximates the functionh(k) by polynomials instead of interpolat-ing the whole integrand in (2.2). When h(k) is continuous the accuracy of the

approxi-mation is guaranteed by the Weierstrass approxiapproxi-mation theorem (seeJeffreys and Jeffreys

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P(k)|< , then I−Q = b a eimkh(k) dk− b a eimkP (k) dk = b a eimk[h(k)−P (k)]dk ≤ b a |eimk[h(k)−P (k)]|dk ≤ b a |h(k)−P (k)|dk ≤ |b−a|max k |h(k)−P (k)|.

The last two lines hold since |eimk| is always less or equal to 1. Hence the error of the

Filon quadrature is bounded as follows:

error(I−Q)≤ |b−a|max

k |h(k)−P (k)|,

which goes to zero if P (k) is an excellent approximation to h(k). From the above error estimation, the accuracy of Filon quadrature is free from the oscillatory factor.

Following Ballestra et al. (2007) we apply a Filon-type quadrature to our problem. Before doing that we change the integral in (2.1) as follows:

+∞ 0

eimkh(k) dk =

+∞ 0

[cos(mk) +isin(mk)]×[Re{h(k)}+iIm{h(k)}] dk

= +∞ 0 cos(mk)Re{h(k)} −sin(mk)Im{h(k)} dk + +∞ 0 i[cos(mk)Im{h(k)}+sin(mk)Re{h(k)}] dk = +∞ 0 cos(mk)Re{h(k)} −sin(mk)Im{h(k)} dk.

The last equation holds since the integral is a probability density function which must be real. The upper bound of the integral +∞ can be replaced by a constant kmax since the

absolute value of the integrand goes to zero as k increases. Hence, + 0 eimkh(k) dk ≈ kmax 0 eimkh(k) dk.

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It is easy to choose a kmax such that |eimkh(k)| < for all possible xt and vt, given the

Heston parameters,x0 and v0. In this paper, we choose kmax = 60 such that

|eimkh(k)|< = 10−5,

when k > kmax.

In what follows, we illustrate how we can interpolate h(k). Our idea is to use piece-wise interpolation. We first divide the interval [0, kmax] into M subintervals at points x0, x1, . . . , xM, then we interpolate h(k) on each interval. In order to have efficient

com-putation time, we use 6th-order Lagrange interpolation in each subinterval, and the inter-polation points in each subinterval are a linear transformation of the roots of the Legendre polynomial of degree 7. In the ith subinterval, the interpolation points are denoted by

xi 0, x

i

1, . . . , x i

6. InFilon(1928) the interpolation was a quadratic spline, the spline could be

more accurate but it costs more time since the derivatives of h(k) and the interpolation function at boundaries points of each subinterval are set to be the same for spline. We denote the piece-wise Lagrange interpolation as L(k). Using this method the integral

(2.2) becomes

kmax

0

eimkL(k) dk (2.3)

and it can be solved analytically sinceL(k) is a polynomial. The quadrature error depends on the Lagrange interpolation, that is, for anyk in [a, b]

error(h(k)−L(k))≤max i maxx∗i f(n+1)(ξi) (n+ 1)! ωi(x ∗ i) , whereξi, x∗i ∈[xi−1, xi] and ωi(x) = x−xi0 x−xi1. . .x−xi6.

Hence, the quadrature error is bounded by

kmaxmax i maxx∗i f(n+1)(ξi) (n+ 1)! ωi(x ∗ i) .

Here, we set n = 6 which is large enough. The Filon-type quadrature is faster since all the computations are evaluated analytically except for the modified Bessel function in (2.2).

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2.3

Pricing European Path-Dependent Derivatives using TPDF

It is challenging to price path-dependent derivatives in the Heston framework. Although they can be priced by simulating the asset price, simulation methods have some drawback-s. It is hard to evaluate and eliminate the discretization error. Another shortcoming is that it is hard to simulate the Greeks directly, which represent the sensitivities of the price with respect to financial market changes. In this subsection we show how to price Euro-pean pathdependent derivatives by JTPDFs, a method which overcomes the simulation’s shortcoming.

Since Heston’s process is a Markovian process ofxtandvt, given the initial log-pricex0

and stochastic variancev0, the joint PDF of x= (x1, x2, . . . , xn) and v = (v1, v2, . . . , vn)

can be written as:

p(x,v|x0, v0) =

n

"

t=1

p(xt, vt|xt−1, vt−1). (2.4)

Using (2.4) it is possible to price European path-dependent derivatives analytically.

Let S= (S0, S1, . . . , Sn), we denote the payoff of a European path-dependent

deriva-tive byCpath(S, n). Note thatSi =Si(xi) =exi, the payoff is rewritten asCpath(S(x), n),

whereS(x) = (S0(x0), S1(x1), . . . , Sn(xn)). Note that the time unit represents one

peri-od, so that we let the initial timet0 be 0 but it can represent any time before maturity n.

Then, according to Proposition1.3, the price of the European path-dependent derivative,

Πpath(0, n, x0, v0), is the expectation of its discounted payoff, that is,

Πpath(0, n, x0, v0) = EQ e−0nrwdwC path(S(x), n) x0, v0 = e−0nrwdwEQ C path(S(x), n) x0, v0 = e−0nrwdw · · · (x,v)∈Ω Cpath(S(x), n) n " j=1 p(xj, vj|xj−1, vj−1) dxdv, (2.5) where Ω = [−∞,+∞]n×[0,+∞]n.

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Note that (2.5) could be solved recursively by tacking

(xn,vn)∈[−∞,+∞]×[0,+∞]

Cpath(S(x), n)p(xn, vn|xn−1, vn−1) dxndvn,

for givenxn−1 andvn−1. Hence, it becomes possible to solve (2.5). Lipton(2001) first used

this idea to price the forward starting option. However, while n gets large it is rather challenging to solve the multidimensional integral. Ballestra et al. (2007) introduced a numerical method to price the path-dependent derivatives and they succeed to price an one-year arithmetic Asian option. Because the main purpose of our paper is to price a 7-year EIA contract, which means that (2.5) is a 20-dimensional integral, we solve the

multiple integral by a numerical method.

2.4

Importance Sampling

2.4.1 Importance Sampling

FollowingBallestra et al.(2007), we solve the multiple integral in (2.5) using the method of importance sampling. In this case the standard Monte Carlo method is inefficien-t inefficien-to evaluainefficien-te inefficien-the expecinefficien-tainefficien-tion since iinefficien-t is difficulinefficien-t inefficien-to generainefficien-te samples from inefficien-the densiinefficien-ty

p(xj, vj|xj−1, vj−1). Thus the importance sampling method is introduced to overcome

this problem.

The idea of importance sampling is as follows. Instead of generate samples from

p(xj, vj|xj−1, vj−1), we first approximate p(xj, vj|xj−1, vj−1) by another similar function

˜

p(xj, vj|xj−1, vj−1) which must be 0 wherep(xj, vj|xj−1, vj−1) = 0. Then (2.5) becomes:

Πpath(0, n, x0, v0) = e−0nrwdw · · · (x,v)∈Ω Cpath(S(x), n)× #n j=1p(xj, vj|xj−1, vj−1) #n j=1p˜(xj, vj|xj−1, vj−1) n " j=1 ˜ p(xj, vj|xj−1, vj−1) dxdv = e−0nrwdwEQ ˜ p Cpath(S(x), n) #n j=1p(xj, vj|xj−1, vj−1) #n j=1p˜(xj, vj|xj−1, vj−1) x0, v0 , (2.6)

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whereEpQ˜ [·] is the expectation under Q w.r.t

#n

j=1p˜(xj, vj|xj−1, vj−1).

Hence, to evaluate the multiple integral in (2.5), we just need to generate samples

from ˜p(xj, vj|xj−1, vj−1). Hence, what is left is to find an appropriation ˜p whose samples

are easy to generate.

2.4.2 JTPDF Approximation

Again following Ballestra et al. (2007), ˜p(xj, vj|xj−1, vj−1) is given by bilinear

interpola-tion. This is because samples of a density described by a bilinear interpolation function are easy to generate. An introduction to bilinear interpolation is given in Appendix D.

In terms of approximating the JTPDFp(xt+τ, vt+τ|xt, vt), note that in (2.1) bothxt+τ

andxtalways appear together in the term ofxt+τ−xt, so that (2.1) is actually a function

of vt+τ, vt and Δxt =xt+τ −xt. Hence, the JTPDF is a function of three variables, p(xt+τ, vt+τ|xt, vt) =p(Δxt, vt+τ|0, vt). (2.7)

In fact, (2.7) shows that, given vt, the process for xt is time homogeneous. Since the

support is rather small compared to its domain, we focus on approximating p on its support.

Assume thatvtis given, we can find an upper boundvmaxforvt+τ using the transition

PDF in (1.2), such that p(vt+τ|vt)< εv when vt+τ is outside the interval [0, vmax].

In terms of Δxt, note that the process is described as dxt= rt− 1 2vt dt+√vtdWsQ(t).

We discretize the previous process as Δxt ≈ t+τ t r(w) dw− 1 2(σ ∗)2 τ+σ∗Z,

where Z ∼ N (0, τ). In terms of σ∗, for vt is a mean-reverting process it makes sense

to set σ∗ = E[vt+τ|vt]. Then the PDF of Δxt is estimated by a normal distribution Ntt+τr(w)dw−21 (σ∗)2τ,(σ∗√τ)2

. Thus, the support of Δxt is estimated by

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[Δmin,Δmax] t+τ t r(w)dw− 1 2(σ ∗)2 τ −αxσ √ τ , t+τ t r(w) dw− 1 2(σ ∗)2 τ −αxσ √ τ

where αx is a number such that the probability for Δxt to be outside the above interval

is smaller than εx. Hence [0, vmax]×[Δmin,Δmax] is used as the support of the TPDF, p{Δxt, vt+τ|0, vt}.

Divide the above support intoNv×Nx grids. Then the look-up table can be generated

using the Filon quadrature. Finally, p(Δxt, vt+τ|0, vt), given vt, can be approximated by

a bilinear interpolation ˜p(Δxt, vt+τ|0, vt).

In the above, vt is assumed to be known while in practice it is changing from time

to time. Hence, p(Δxt, vt+τ|0, vt) is also a function of vt. Suppose that vt+τ and Δxt

are given, we use piece-wise interpolation to approximate JTPDF as a function of vt.

Since the transition density function is given in (1.2) and the process vt has a

mean-reverting property, we can find an interval [0, μmax] for vt, given any possible vt−τ, such

that p(vt|vt−τ) < εv when vt is outside [0, μmax]. Hence, vt must be in the interval

[0, μmax]. Divide the interval [0, μmax] into M subintervals and denote the break-points

by {μ1, μ2, . . . , μM, μM+1}, then vt must fall into one of the subintervals. Sayvt falls into

the interval [μi−1, μi], given Δxt andvt+τ, then the approximation ofp{Δxt, vt+τ|0, vt}is

given by ˜ p(Δxt, vt+τ|0, vt)≈ μi−vt μi−μi−1 ˜ p(Δxt, vt+τ|0, μi−1) + vt−μi−1 μi−μi−1 ˜ p(Δxt, vt+τ|0, μi), (2.8)

whereui, i= 1, . . . , M + 1, are already known.

This leads to a good approximation because vmax cannot be too large since the initial

volatility is usually smaller than 0.1. Also, the JTPDF is a continuous function ofvt and

if M is big enough the error remains acceptable.

In summary, the algorithm for approximating the JTPDF is given in Algorithm 1

Figure

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References

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